Optimal. Leaf size=98 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}} \]
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Rubi [A] time = 0.0332707, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {442} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 442
Rubi steps
\begin{align*} \int \frac{x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0589936, size = 70, normalized size = 0.71 \[ -\frac{x^3 \left (\frac{a+b x^2}{a}\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )}{6 a \left (-a-b x^2\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-b{x}^{2}-2\,a} \left ( -b{x}^{2}-a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2}}{{\left (b x^{2} + 2 \, a\right )}{\left (-b x^{2} - a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67294, size = 559, normalized size = 5.7 \begin{align*} 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \arctan \left (\frac{4 \,{\left (\sqrt{\frac{1}{2}} \left (\frac{1}{4}\right )^{\frac{3}{4}} a b^{4} x \sqrt{\frac{b^{4} x^{2} \sqrt{\frac{1}{a b^{6}}} + 2 \, \sqrt{-b x^{2} - a}}{x^{2}}} \left (\frac{1}{a b^{6}}\right )^{\frac{3}{4}} - \left (\frac{1}{4}\right )^{\frac{3}{4}}{\left (-b x^{2} - a\right )}^{\frac{1}{4}} a b^{4} \left (\frac{1}{a b^{6}}\right )^{\frac{3}{4}}\right )}}{x}\right ) - \frac{1}{2} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \log \left (\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} +{\left (-b x^{2} - a\right )}^{\frac{1}{4}}}{x}\right ) + \frac{1}{2} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \log \left (-\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (\frac{1}{a b^{6}}\right )^{\frac{1}{4}} -{\left (-b x^{2} - a\right )}^{\frac{1}{4}}}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{2 a \left (- a - b x^{2}\right )^{\frac{3}{4}} + b x^{2} \left (- a - b x^{2}\right )^{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2}}{{\left (b x^{2} + 2 \, a\right )}{\left (-b x^{2} - a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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